568 ON THE LIGHT EMITTED FKOM A [436
It will be observed that for m > 2, dR/dsc ~ 0 when as = 0, but that for m = 2, dR/doo - 4.
The corresponding question for J may be worth a moment's notice. We have
.v,., 7m— 1
tf-W'W-sfe-,; ........................ (8)
so that R' goes to zero as m increases, if / be comparable with n, as might have 'been expected.
It must not be overlooked that when the random distribution of phases is due to a random spatial distribution of centres, it fails to satisfy strictly the requirement that all the centres act independently, for some of them will lie at distances from nearest neighbours less than the number of wave-lengths necessary for approximate independence. The simple conditions just discussed are thus an ideal, approached only when the spacing is very open.
We have now to consider how the question is affected when we abandon the restriction that the spacing of the unit centres is very open. The work to be clone at each centre then depends not only upon the pressure due to ibself but also upon that due to not too distant neighbours. Beginning with a single source, we may take as the velocity-potential
where a is the velocity of propagation, Js = 27T/X, and r is the distance from the centre. The rate of passage of fluid across the sphere of radius r is
4tTrr^d<^/dr = cos Jc (at — r} — Jcr sin k (at — r) ............. (10)
If $p denote the variable part of the pressure at the same time and place,
and p be the density,
&n-_0-i __ pkasmk(at-r)
dp~ Pdt~~ ~~4nrr ............. (1Ua)
The rate at which work ( W) has to be done is given by
„ . „ d<t> pica, sin k (at — r}
= S . 4TTT- — - = ~ - ----- -N— —'-
j . — - - ----- -
at r dr 477T
x [kr sin k (at — r) — cos k (at — r)], . . .(10 b)*
of which the mean value depends upon the first term only. In the long run
Wft**pfra/bir ......................... (10 c)*
It is to be observed that although the pressure is infinite at the source, the work done there is nevertheless finite on account, of the pressure being in quadrature with the principal part of the rate of total flow expressed in (10).
[* In the original paper these equations "were numbered (11) — (13), as well as the three following equations ; to avoid confusion they have been renumbered .]s2 6, reducing to G simply, if A = G. This is the result for a single particle whose axis is at W. What we are aiming